Semiparametric methods for left-truncated and right-censored survival data with covariate measurement error

Abstract

Many methods have been developed for analyzing survival data which are commonly right-censored. These methods, however, are challenged by complex features pertinent to the data collection as well as the nature of data themselves. Typically, biased samples caused by left-truncation (or length-biased sampling) and measurement error often accompany survival analysis. While such data frequently arise in practice, little work has been available to simultaneously address these features. In this paper, we explore valid inference methods for handling left-truncated and right-censored survival data with measurement error under the widely used Cox model. We first exploit a flexible estimator for the survival model parameters which does not require specification of the baseline hazard function. To improve the efficiency, we further develop an augmented nonparametric maximum likelihood estimator. We establish asymptotic results and examine the efficiency and robustness issues for the proposed estimators. The proposed methods enjoy appealing features that the distributions of the covariates and of the truncation times are left unspecified. Numerical studies are reported to assess the finite sample performance of the proposed methods.

Appendix: Regularity conditions

Like any other asymptotic results, the validity of our results requires regularity conditions imposed on the processes of survival, censoring, measurement error and covariates as well as the sampling scheme. Basically, our regularity conditions pertain to those in Andersen and Gill (1982), Huang et al. (2012), and Yan and Yi (2016), including the following assumptions:

• (C1) $\varTheta$ is a compact set, and the true parameter value $\beta _0$ is an interior point of $\varTheta$.

• (C2) $\int _0^\tau \lambda _0(t)\mathrm{d}t < \infty$, where $\tau$ is the finite maximum support of the failure time.

• (C3) The $\left\{ N_i(t), Y_i(t),Z_i,X_i \right\}$ are independent and identically distributed for $i=1,\ldots ,n$.

• (C4) The covariates $Z_i$ and $X_i$ are bounded.

• (C5) Conditional on $V_i^*$, $\left( T_i^*, V_i^*\right)$ are independent of $A_i^*$.

• (C6) Censoring time $C_i$ is non-informative. That is, the failure time $T_i$ and the censoring time $C_i$ are independent, given the covariates $\{Z_i, X_i\}$.

• (C7) Matrices $E \left( -\frac{1}{n} \frac{\partial ^2 \ell _C^*}{\partial \beta \partial \beta ^\top } \right)$ and $E \left( - \frac{1}{n}\frac{\partial ^2 \ell _M^*}{\partial \beta \partial \beta ^\top } \right)$ are positive definite, where $\ell _C^*$ is defined in (10) and $\ell _M^*$ is the logarithm of the likelihood function (19).

• (C8) The operations of differentiation and integration are exchangeable.

Condition (C1) is a basic condition that is used to derive the maximizer of the target function (e.g., Huang et al. 2012, p.203). (C2) to (C6) are standard conditions for survival analysis, which allow us to obtain the sum of independent and identically distributed random variables and hence to derive the asymptotic properties of the estimators (e.g., Andersen and Gill 1982). The requirement of positive definite matrices in Condition (C7) is standard which ensures asymptotic covariance matrices of $\ell _C^*$ and $\ell _M^*$ meaningful. Condition (C8) is a routine requirement for deriving asymptotic results.

Lemma 1

Let

$$ {\widehat{\ell }}_P^*= \sum _{i=1}^{n} \int \nolimits _{0}^{\tau } \left[ {\widetilde{v}}_i^\top \beta + \frac{1}{2} \beta _{x}^\top {\varSigma_{\epsilon}} \beta _{x} - \log \left\{ \sum _{j=1}^{n} \exp ({\widetilde{v}}_j^\top \beta ) I(a_j \le u \le y_j) \right\} \right] \mathrm{d}N_i (u). $$

(40)

Then (10) and (40) yield the same maximum likelihood estimator of $\beta$.

The proof is given in Appendix B of the Supplementary Material. The following lemma is used to establish the consistency of the estimators ${\widehat{\beta }}$ and ${\widetilde{\beta }}$, respectively, given in Theorems 2 and 3.

Lemma 2

Define

$$\begin{aligned} \kappa _P = E\left( \frac{1}{n} {\widehat{\ell }}_P^*\right) \end{aligned}$$

and let

$$ \kappa= \kappa _P + E\left\{ \frac{1}{n} \log \left( L_M^*\right) \right\} , $$

where ${\widehat{\ell }}_P^*$ and $L_M^*$ are determined by (40) and (19), respectively, with the data $\{ {\widetilde{v}}_i, a_i, y_i,z_i \}$ replaced by the corresponding random variables $\{ {\widetilde{V}}_i, A_i, Y_i,Z_i \}$. Then $\beta _0$ is the unique maximizer of $\kappa _P$ and $\kappa$.

Proof

Part 1: We show that $\beta _0$ is the unique maximizer of $\kappa _P$.

Recall that $\ell _C$ is the logarithm of the likelihood function (2) based on the true covariates X. In the absence of measurement error, i.e., based on the true covariates X, Huang et al. (2012, p.208) showed that the true value $\beta _0$ is the unique maximizer of $E(\ell _C)$. Noting that by (9), $\ell _C$ and $\ell _C^*$, defined in (10), have the relationship

$$\begin{aligned} E(\ell _C^*) = E(\ell _C). \end{aligned}$$

(41)

We conclude that $\beta _0$ is also the unique maximizer of $E(\ell _C^*)$. By Lemma 1, we conclude that $\beta _0$ is the unique maximizer of $\kappa _P$. With regularity conditions including (C8),

$$\begin{aligned} \beta _0 \ \ \text {is the unique solution of} \ \ E\left( \frac{1}{n} \frac{\partial {\widehat{\ell }}_P^*}{\partial \beta } \right) = 0, \end{aligned}$$

(42)

and

$$\begin{aligned} \left. \frac{\partial ^2 \kappa _P}{\partial \beta \partial \beta ^\top } \right| _{\beta =\beta _0} \ \ \text {is negative definite.} \end{aligned}$$

(43)

Part 2: We show that $\beta _0$ is the unique maximizer of $\kappa$.

Let $\ell _M(\beta ;X,Z)$ denote the logarithm of the likelihood function (3) based on the true covariates X, and let $\ell _M(\beta ;{\widetilde{X}},Z)$ be $\ell _M(\beta ;X,Z)$ with X replaced by ${\widetilde{X}} = E\left( X | W^*\right)$, where $W^*= W-{\varSigma_{\epsilon}} \alpha$ as defined before (7). Define $U_M(\beta ;X,Z) = \frac{\partial \ell _M(\beta ;X,Z)}{\partial \beta }$ and let ${\mathcal{U}}_M(\beta ;X,Z) = E\left\{ {\frac{1}{n}} U_M(\beta ;X,Z) \right\}$.

Recall that $\mu _X = E(X)$ defined before (17), then by (7), we have that $E({\widetilde{X}}) = \mu _X$. Let $\mu _Z = E(Z)$. Then by the linear approximation around $\mu _X$ and $\mu _Z$, we express $U_M(\beta ;{\widetilde{X}},Z)$ and $U_M(\beta ;X,Z)$, respectively, as

$$\begin{aligned} U_M(\beta ;{\widetilde{X}},Z)& {} \approx U_M(\beta ;\mu _X,\mu _Z) + \frac{\partial U_M(\beta ;\mu _X,\mu _Z)}{\partial \mu _X} ({\widetilde{X}} - \mu _X) \nonumber \\&\quad + \frac{\partial U_M(\beta ;\mu _X,\mu _Z)}{\partial \mu _Z} (Z - \mu _Z) \nonumber \\ \end{aligned}$$

(44)

and

$$\begin{aligned} U_M(\beta ;X,Z)& {} \approx U_M(\beta ;\mu _X,\mu _Z) + \frac{\partial U_M(\beta ;\mu _X,\mu _Z)}{\partial \mu _X} (X - \mu _X)\nonumber \\&\quad + \frac{\partial U_M(\beta ;\mu _X,\mu _Z)}{\partial \mu _Z} (Z - \mu _Z), \nonumber \\ \end{aligned}$$

(45)

where $\frac{\partial U_M(\beta ;\mu _X,\mu _Z)}{\partial \mu _X}$ represents the partial derivative $\frac{\partial U_M(\beta ;a,b)}{\partial a}$ evaluated at $(a,b) = (\mu _X,\mu _Z)$, and $\frac{\partial U_M(\beta ;\mu _X,\mu _Z)}{\partial \mu _Z}$ represents the partial derivative $\frac{\partial U_M(\beta ;a,b)}{\partial b}$ evaluated at $(a,b) = (\mu _X,\mu _Z)$. Here $U_M(\beta ;a,b)$ has the same functional form as $U_M(\beta ;X,Z)$ except that the former is a real-valued function with arguments $\beta$, a and b, while the latter case is a function of random variables X and Z together with $\beta$.

Combining (44) and (45) gives that

$$\begin{aligned} U_M(\beta ;{\widetilde{X}},Z) \approx U_M(\beta; X,Z) + \frac{\partial U_M(\beta ;\mu _X,\mu _Z)}{\partial \mu _X} ({\widetilde{X}} - X). \end{aligned}$$

(46)

Therefore, taking expectation on both sides of (46) and replacing $\beta$ by $\beta _0$ give

$$\begin{aligned} {\mathcal{U}}_M(\beta _0; {\widetilde{X}},Z) \approx 0 \end{aligned}$$

(47)

because that $E( {\widetilde{X}}) - E(X) = \mu _X - \mu _X = 0$ and ${\mathcal{U}}_M(\beta _0; X,Z) = 0$ (e.g., Huang et al. 2012, p.208).

By (42) and (47),

$$\begin{aligned} E\left( \left. \frac{1}{n} \frac{\partial {\widehat{\ell }}_P^*}{\partial \beta } \right| _{\beta =\beta _0} \right) + {\mathcal{U}}_M(\beta _0; {\widetilde{X}},Z)\approx 0. \end{aligned}$$

(48)

By definition of $U_M(\beta ;X,Z)$ and (19) together with (17), we have $U_M(\beta ;{\widetilde{X}},Z) = \frac{\partial \log (L_M^*)}{\partial \beta }$, and thus, ${\mathcal{U}}_M(\beta ;{\widetilde{X}},Z) = E\left\{ \frac{1}{n} \frac{\partial \log (L_M^*)}{\partial \beta } \right\}$ and $\frac{\partial {\mathcal{U}}_M(\beta ;{\widetilde{X}},Z)}{\partial \beta } = E\left\{ \frac{1}{n} \frac{\partial ^2 \log (L_M^*)}{\partial \beta \partial \beta ^\top } \right\}$. Then applying (48) gives

$$\begin{aligned} \left. \frac{\partial \kappa }{\partial \beta } \right| _{\beta =\beta _0} \approx 0. \end{aligned}$$

(49)

Next, by taking expectation on (46) and then taking the partial derivative with respect to $\beta$ give that

$$\begin{aligned} \frac{\partial {\mathcal{U}}_M(\beta ;{\widetilde{X}},Z)}{\partial \beta } \approx \frac{\partial {\mathcal{U}}_M(\beta ;X,Z)}{\partial \beta }. \end{aligned}$$

(50)

By the derivations similar to Huang et al. (2012, p.208), $\frac{\partial {\mathcal{U}}_M(\beta ;X,Z)}{\partial \beta }$ is negative definite at $\beta =\beta _0$, and thus, by (50), $\frac{\partial {\mathcal{U}}_M(\beta ;{\widetilde{X}},Z)}{\partial \beta }$ is also negative definite at $\beta =\beta _0$. Then combining with (43) gives that $\frac{\partial ^2 \kappa }{\partial \beta \partial \beta ^\top } = \frac{\partial ^2 \kappa _P}{\partial \beta \partial \beta ^\top } + E\left\{ \frac{1}{n} \frac{\partial ^2 \log (L_M^*)}{\partial \beta \partial \beta ^\top } \right\}$ is negative definite at $\beta =\beta _0$. Therefore, combining with (49), we conclude that $\beta _0$ is approximately the maximizer of $\kappa$. $\square$